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Let $[x]$ be the floor function, i.e., $[x]$ denotes the largest integer that is not bigger than $x$. Calculate the double integral:

$$\iint_R [x+y] dxdy$$

...where $R=\{(x,y): 0\leq x\leq 2, 0\leq y\leq 2\}.$

A

$0$

B

$-1$

C

$2$

D

$4$

E

$6$

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